Osculating Orbits and Local Self-Consistent Evolution

An accelerated EMRI worldline is not a Kerr geodesic, but at each event it can be represented by the unique member of a suitably parameterized geodesic family with the same position and tangent. The resulting osculating elements turn a local self-force into evolution equations for orbital constants and phases. Osculation is a change of variables, not an adiabatic approximation by itself.

Helpful background. The Einstein Equation fixes the background field equation, Weak Metric Perturbations and Gauge Freedom supplies perturbative gauge control, and Presymplectic Potentials and Currents clarifies conserved quantities. Black-Hole Perturbation Theory supplies the regular field, Relativistic Two-Body Dynamics supplies the mass-ratio expansion, and Gravitational Radiation and Waveforms supplies balance checks.

Osculation as a tangent-space condition

Let the Kerr black hole have mass $M_\bullet$, the secondary have mass $m_\star$, and $\epsilon=m_\star/M_\bullet\ll1$. The orbital elements below are specific quantities, with factors of $m_\star$ divided out.

Let a family of Kerr geodesics be written

$$z_{\mathrm G}^\mu(\tau;I^A),$$

where the elements $I^A$ include constants such as $(E,L_z,Q)$ and enough phase offsets to specify a point on the orbit. Represent an accelerated worldline by

$$z^\mu(\tau)=z_{\mathrm G}^\mu(\tau;I^A(\tau)).$$

Equality of positions is not sufficient. The geodesic and accelerated worldlines must also have the same tangent:

$$u^\mu(\tau) =u_{\mathrm G}^\mu(\tau;I^A(\tau)).$$

Differentiating the position representation gives the osculation constraints

$$\frac{\partial z_{\mathrm G}^\mu}{\partial I^A} \frac{dI^A}{d\tau}=0.$$

Differentiating the tangent covariantly then gives

$$a^\mu =\frac{\partial u_{\mathrm G}^\mu}{\partial I^A} \frac{dI^A}{d\tau},$$

because the geodesic acceleration vanishes when the elements are fixed. These two linear systems determine the element rates from the applied acceleration, subject to the chosen parameterization and constraints. Redundant orbital elements make the system singular; omitting the phase elements generally makes the position and tangent conditions incompatible.

For a force that preserves the small body’s rest mass,

$$u_\mu a^\mu=0.$$

This orthogonality is an inexpensive diagnostic of a projected self-force and of the numerical element inversion.

Direct evolution of Kerr integrals

The rates of the geodesic integrals follow without selecting $(p,e,\iota)$. With stationary and axial Killing vectors $\xi_{(t)}^\mu$ and $\xi_{(\phi)}^\mu$,

$$\frac{dE}{d\tau} =-\xi^{(t)}_\mu a^\mu, \qquad \frac{dL_z}{d\tau} =\xi^{(\phi)}_\mu a^\mu.$$

Let $K=K_{\mu\nu}u^\mu u^\nu$ be the Killing-tensor integral and use the same Carter constant as on the Kerr geodesics page, $Q=K-(L_z-aE)^2$, with fixed background spin parameter $a=J_\bullet/M_\bullet$. The Killing-tensor identity $\nabla_{(\rho}K_{\mu\nu)}=0$ gives

$$\frac{dK}{d\tau} =2K_{\mu\nu}u^\mu a^\nu.$$

Differentiating the subtraction therefore gives

$$\frac{dQ}{d\tau} =2K_{\mu\nu}u^\mu a^\nu -2(L_z-aE)\left(\frac{dL_z}{d\tau} -a\frac{dE}{d\tau}\right).$$

This distinction matters even for equatorial motion: reflection symmetry preserves $Q=0$, while $K=(L_z-aE)^2$ generally evolves. Coordinate-time rates are $dI^A/dt=(dI^A/d\tau)/u^t$; the time conversion belongs inside an orbit average when $u^t$ varies along an eccentric orbit.

These formulas determine the three principal elements. The osculation conditions supply the accompanying radial, polar, temporal, and azimuthal phase evolution. A Jacobian then maps $(\dot E,\dot L_z,\dot Q)$ to $(\dot p,\dot e,\dot\iota)$ where that map is regular.

Near a turning point, formulas that divide by $u^r$ or $u^\theta$ can appear singular even though the physical acceleration is finite. A robust implementation changes variables, takes the analytic turning-point limit, or uses a formulation whose independent equations remain regular. Near a separatrix, the map from constants to orbital parameters becomes ill conditioned for a physical reason: a small change in constants can produce a large change in radial period.

The general forced-geodesic and osculating constructions, including Kerr implementations, are developed in Gair et al. 2011, §§II–IV and reviewed in Pound and Wardell 2021, §§6.2.2–6.2.6, pp. 65–84.

Three evolution strategies

The phrase “forced geodesic” covers distinct approximations.

Osculating evolution. The exact acceleration model is evaluated and translated into element rates continuously. If the acceleration is exact and the element map is nonsingular, this is simply an equivalent representation of the accelerated equation of motion.

Geodesic self-force evolution. The self-force is computed for a fixed geodesic, and its tabulated values are used while the elements drift. Updating the table continuously or by interpolation can be accurate through a declared order, but the field source is not the evolving accelerated worldline.

Self-consistent evolution. The regular field and force are evolved with the actual accelerated source worldline. This avoids a long-time expansion about one reference geodesic but is substantially more expensive. At fixed perturbative order it still carries gauge, regularization, initial-data, and truncation errors; “self-consistent” does not mean exact.

An osculating parameterization can be used in either of the last two schemes. It therefore should not be contrasted with self-consistency as though they were mutually exclusive physical models.

Averaging and an independent balance check

Decompose the local first-order acceleration into conservative and dissipative pieces and into torus-average and oscillatory parts. Away from resonance, averaging the osculating rates gives

$$\left\langle\dot I^A_{\mathrm{osc}}\right\rangle =\left\langle\dot I^A_{\mathrm{flux}}\right\rangle +O(\epsilon^2)$$

for the dissipative actions, after time parameters and normalizations are matched. Energy and axial angular momentum compare directly with horizon-plus-infinity fluxes. Carter evolution compares with its radiative self-force formula, not with a fictitious asymptotic Carter flux.

This equality tests the local-force projection, orbit averaging, mode normalization, and sign convention simultaneously. It should fail in a predictable way at resonance: the long-time torus average discards the phase-dependent harmonic that becomes stationary. It also ceases to be the complete description near the plunge transition.

Setting $a^\mu=0$ provides the simplest check. The element equations must give $\dot I^A=0$, while the geodesic phases continue to advance at their fundamental frequencies. If the orbital constants drift in this limit, the osculation constraints or their numerical inversion are wrong.

Exercises

Derive the rate of the specific energy from the stationary Killing vector.

Solution

With $E=-\xi^{(t)}_\mu u^\mu$,

$$\frac{dE}{d\tau} =-u^\nu\nabla_\nu(\xi^{(t)}_\mu u^\mu) =-\xi^{(t)}_\mu a^\mu -u^\nu u^\mu\nabla_\nu\xi^{(t)}_\mu.$$

The last term vanishes because $u^\nu u^\mu$ is symmetric while $\nabla_{(\nu}\xi^{(t)}_{\mu)}=0$. Thus $\dot E=-\xi^{(t)}_\mu a^\mu$, and it vanishes for geodesic motion.

Why are phase elements required even if only $E$, $L_z$, and $Q$ are of long-term interest?

Solution

The three constants select a geodesic torus but not the point on that torus. An accelerated worldline must match both position and tangent at each instant, which requires radial and polar phase offsets and the corresponding temporal and azimuthal origins. Without them, the element family lacks enough freedom to satisfy the osculation constraints.

References

• Gair, Jonathan R., Éanna É. Flanagan, Steve Drasco, Tanja Hinderer, and Stanislav Babak. “Forced Motion Near Black Holes.” Physical Review D 83 (2011): 044037. DOI.
• Pound, Adam, and Barry Wardell. “Black Hole Perturbation Theory and Gravitational Self-Force.” In Handbook of Gravitational Wave Astronomy, edited by Cosimo Bambi, Stavros Katsanevas, and Konstantinos D. Kokkotas, 1–119. Singapore: Springer (2021). DOI.